Stability Space and the Below-n Threshold

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Abstract

Stability in tonal and early-atonal music is associated with
concepts of “force” as defined by Larson (1994), Lerdahl and
Jackendoff (1983), Lerdahl (1994, 2001), Schoenberg (Cherlin
2000), Baroni (1983), and Arnheim (Brower 2000). This paper
seeks to frame discussions on stability by constructing a notion of
stability space – a nexus of the time-space (T) and pitch-space
(Π) in which the forces causing stability and instability occur.
Because “force” cannot be delimited by a clear hierarchical
structuring mechanism as proposed by Schenker (1935) and
Lerdahl and Jackendoff (1983), musical moments can belong to
multiple stability spaces of varying temporal and pitch
dimensions. The below-n threshold (B) is an algorithm that finds
trends between these dimensions by discerning the maximum-sized T
containing Π of average cardinality n in various works.
Because this divisional scheme necessarily cuts through events and
event collections, it does not comport with traditional musical
segmentation procedures as defined by Hanninen (2001). Rather, it
uses an approach suggested by Mandelbrolt (1964) in analyzing the
correlation between measurement (|Π|) and the unit of
measurement (T).

A study of Scriabin’s late preludes, for example, find a drastic
change in the value of between his extended-diatonic and octatonic
pitch languages. The data suggests that, beyond simply adopting a
scale with one more pc, Scriabin’s mature style also accessed a
more ambitious chromatic palette. Such an assertion is
supported by Chang (2006), Callender (1998), Kim (1994), and
Meeks (1945).

A second study into Schoenberg’s op. 23 Klavierstücke and op. 25
Suite shows divergence in the below-n threshold after n=9,
suggesting that the two collections maintain a similar pitch
practice in all but the largest pc sets (Example 3). This claim is
supported by Hamao (1988) and Hyde (1985). A musical
justification for the divergence between n=9 and n=10 is
suggested by Hyde, who identifies serial practices that involve
pitch sets of cardinality 8-10 in at least three op. 23
stücke.

Through these case studies, I hope to show how a relatively basic
conceptualization of the pitch and time domain can reveal stable and
instable uses of pitch that are not suggested by conventional means
of musical segmentation.